### Upcoming Talks

#### Zahlentheoretisches Kolloquium

**Title:**Toward a variant of Skolem problem

**Speaker:**Dr. Armand Noubissie (University of Salzburg)

**Date:**08.11.2024, 14 bis 15 Uhr

**Room:**Seminarraum Analysis-Zahlentheorie, Kopernikusgasse 24, 2.OG

**Abstract:**

The Skolem Problem asks to determine whether a given integer linear recurrence sequence has a zero term. This problem, whose decidability has been open for 90 years , arises across a wide range of topics in computer science and dynamical system. In 1977, A generalization of this problem was made by Loxton and Van der Poorten who conjectured that for any $\epsilon >0$ and $\{u_n\}$ a linear recurrence sequence with dominant (s) roots $>1$ in absolute value, there is a effectively computable constant $C(\epsilon),$ such that if $ \vert u_n \vert < (\max_i\{ \vert \alpha_i \vert \})^{n(1-\epsilon)}$, then $n<C(\epsilon)$. Using results of Schmidt and Evertse, a complete non-effective (qualitative) proof of this conjecture was given by Fuchs and Heintze (2021) and, independently, by Karimov and al.~(2023). In this talk, we provide a survey on the study of Universal Skolem set and sketch the proof of the weak version of the conjecture by giving a effective upper bound of the number of solutions of that inequality extending Schmidt and Schlickewei previous works. The higher dimension of this conjecture will also be discussed in this presentation.

Joint work with Florian Luca, James Maynard, Joel Ouaknine and James Worrell.